742edo

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← 741edo

742edo

743edo →

Prime factorization	2 × 7 × 53
Step size	1.61725 ¢
Fifth	434\742 (701.887 ¢) (→ 31\53)
Semitones (A1:m2)	70:56 (113.2 ¢ : 90.57 ¢)
Consistency limit	21
Distinct consistency limit	21

Theory

742edo is a very strong 19-limit system and a zeta peak tuning, and is uniquely consistent in the 21-odd-limit. It tempers out the vishnuzma and the fortune comma in the 5-limit, supporting vishnu and fortune; 2401/2400 in the 7-limit, 9801/9800 in the 11-limit, 4096/4095, 6656/6655, 10648/10647 in the 13-limit, 1701/1700, 2058/2057, 2601/2600, 4914/4913, 5832/5831 in the 17-limit, 2376/2375, 2432/2431, 2926/2925, 3136/3135, 4200/4199, 5776/5775, 5929/5928, 5985/5984, 6860/6859 in the 19-limit.

Prime harmonics

Approximation of prime harmonics in 742edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.000	-0.068	+0.209	-0.093	+0.165	+0.443	+0.166	+0.061	-0.781	+0.611	-0.022
	Relative (%)	+0.0	-4.2	+12.9	-5.7	+10.2	+27.4	+10.3	+3.8	-48.3	+37.8	-1.4
Steps (reduced)		742 (0)	1176 (434)	1723 (239)	2083 (599)	2567 (341)	2746 (520)	3033 (65)	3152 (184)	3356 (388)	3605 (637)	3676 (708)

Subsets and supersets

742 = 2 × 7 × 53, so it notably contains 53edo and 14edo. It supports silicon temperament (224 & 742) with period 14 in the 13-limit, and iodine temperament (159 & 742) with period 53 in the 17-limit.

Regular temperament properties

Subgroup	Comma List	Mapping	Optimal 8ve Stretch (¢)	Tuning Error
				Absolute (¢)	Relative (%)
2.3.5	[23 6 -14⟩, [-84 53⟩	[⟨742 1176 1723]]	-0.0157	0.0555	3.43
2.3.5.7	2401/2400, 14348907/14336000, [23 6 -14⟩	[⟨742 1176 1723 2083]]	-0.0035	0.0525	3.24
2.3.5.7.11	2401/2400, 9801/9800, 172032/171875, 1240029/1239040	[⟨742 1176 1723 2083 2567]]	-0.0123	0.0501	3.10
2.3.5.7.11.13	2401/2400, 4096/4095, 6656/6655, 9801/9800, 39366/39325	[⟨742 1176 1723 2083 2567 2746]]	-0.0302	0.0608	3.76
2.3.5.7.11.13.17	1701/1700, 2058/2057, 2401/2400, 2601/2600, 4096/4095, 6656/6655	[⟨742 1176 1723 2083 2567 2746 3033]]	-0.0317	0.0564	3.49
2.3.5.7.11.13.17.19	1701/1700, 2058/2057, 2376/2375, 2401/2400, 2432/2431, 2601/2600, 3213/3211	[⟨742 1176 1723 2083 2567 2746 3033 3152]]	-0.0295	0.0531	3.28
2.3.5.7.11.13.17.19.23	1197/1196, 1496/1495, 1701/1700, 2025/2024, 2058/2057, 2401/2400, 2601/2600, 3213/3211	[⟨742 1176 1723 2083 2567 2746 3033 3152 3357]] (742i)	-0.0468	0.0699	4.32

• 742et has a lower 19-limit relative error than any previous equal temperaments. It is only bettered by 935 in terms of absolute error, and by 1178 in terms of relative error.
• 742et (742i val) is also notable in the 17- and 23-limit, where it has lower absolute errors than any previous equal temperaments. In the 17-limit it beats 581 and is bettered by 764; in the 23-limit it beats 718 and is bettered by 814.

Rank-2 temperaments

Periods per 8ve	Generator (Reduced)	Cents (Reduced)	Associated Ratio	Temperaments
1	137\742	221.563	8388608/7381125	Fortune
1	303\742	490.026	896/675	Surmarvelpyth
2	44\742	71.159	25/24	Vishnu
14	434\742 (10\742)	701.886 (16.173)	3/2 (105/104)	Silicon
53	239\742 (1\742)	386.523 (1.617)	5/4 (32805/32768)	Mercator
53	565\742 (5\742)	913.746 (8.086)	441/260 (196/195)	Iodine